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We investigate the exact overlaps between eigenstates of integrable spinchains and a special class of states called "integrable initial/final states".These states satisfy a special integrability constraint, and they are closelyrelated to integrable boundary conditions. We derive new algebraic relationsfor the integrable states, which lead to a set of recursion relations for theexact overlaps. We solve these recursion relations and thus we derive newoverlap formulas, valid in the XXX Heisenberg chain and its integrable higherspin generalizations. Afterwards we generalize the integrability condition totwisted boundary conditions, and derive the corresponding exact overlaps.Finally, we embed the integrable states into the "Separation of Variables"framework, and derive an alternative representation for the exact overlaps ofthe XXX chain. Our derivations and proofs are rigorous, and they can form thebasis of future investigations involving more complicated models such as nestedor long-range deformed systems.

On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables